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Weighted Residual Methods

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Computational Techniques for Fluid Dynamics 1

Part of the book series: Springer Series in Computational Physics ((SCIENTCOMP))

Abstract

Weighted residual methods (WRMs) are conceptually different from the finite difference method in that a WRM assumes that the solution can be represented analytically. For example, to obtain the solution of the diffusion equation (3.1) the following approximate solution would be assumed:

$$T=\sum_{j=1}^{J}a_{j}(t)\phi_j(x)$$

where a j(t)are unknown coefficients and ø j (x)are known analytic functions. The terms ø j (x) are often referred to as trial functions and (5.1) as the trial solution. By forcing the analytic behaviour to follow (5.1) some error is introduced unless J is made arbitrarily large. It may be recalled that the finite difference method defines a solution at nodal points only.

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References

  • Baker, A.J. (1983): Finite Element Computational Fluid Mechanics (McGraw-Hill, New York)

    MATH  Google Scholar 

  • Bathe, K.J., Wilson, E.L. (1976): Numerical Methods in Finite Element Analysis (Prentice-Hall, Englewood Cliffs, N.J.)

    MATH  Google Scholar 

  • Bourke, W., McAvaney, B., Puri, K., Thurling, R. (1977): Methods Comput. Phys. 17, 267–325

    MathSciNet  Google Scholar 

  • Brigham, E.O. (1974): The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J.)

    MATH  Google Scholar 

  • Canuto, C., Hussaini, M.Y., Quateroni, A., Zang, T.A. (1987): Spectral Methods in Fluid Dynamics, Springer Ser. Comput. Phys. (Springer, Berlin, Heidelberg)

    Google Scholar 

  • Cooley, J.W., Tukey, J.W. (1965): Math. Comp. 19, 297–301.

    Article  MathSciNet  MATH  Google Scholar 

  • Finlayson, B.A. (1972): The Method of Weighted Residuals and Variational Principles (Academic, New York)

    MATH  Google Scholar 

  • Finlayson, B.A. (1980): Nonlinear Analysis in Chemical Engineering (McGraw-Hill, New York)

    Google Scholar 

  • Fletcher, C.A.J. (1984): Computational Galerkin Methods, Springer Ser. Comput. Phys. (Springer, Berlin, Heidelberg)

    Google Scholar 

  • Fox, L., Parker, I.B. (1968): Chebyshev Polynomials in Numerical Analysis (Oxford University Press, Oxford)

    Google Scholar 

  • Gottlieb, D., Orszag, S.A. (1977): Numerical Analysis of Spectral Methods: Theory and Applications (SIAM, Philadelphia)

    Book  MATH  Google Scholar 

  • Haitiner, G.J., Williams, R.T. (1980): Numerical Prediction and Dynamic Meteorology, 2nd ed. (Wiley, New York)

    Google Scholar 

  • Hussaini, M.Y., Zang, T.A. (1987): Ann. Rev. Fluid Mech 19, 339–367

    Article  ADS  Google Scholar 

  • Isaacson, E., Keller, H.B. (1966): An Analysis of Numerical Methods (Wiley, New York)

    Google Scholar 

  • Jameson, A. (1989): Science 245, 361–371

    Article  ADS  Google Scholar 

  • Jennings, A. (1977): Matrix Computation for Engineers and Scientists (Wiley, Chichester)

    MATH  Google Scholar 

  • Ku, H.C., Hatziavramidis, D. (1984): J. Comput. Phys. 56, 495–512

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ku, H.C., Taylor, T.D., Hirsch, R.S. (1987): Comput. Fluids, 15, 195–214

    Article  ADS  MATH  Google Scholar 

  • Lax, P., Wendroff, B. (1960): Commun. Pure Appl. Math. 13, 217–237

    Article  MathSciNet  MATH  Google Scholar 

  • Mitchell, A.R., Wait, R. (1977): The Finite Element Method in Partial Differential Equations (Wiley, Chichester)

    MATH  Google Scholar 

  • Oden, J.T., Reddy, J.N. (1976): An Introduction to the Mathematical Theory of Finite Elements (Wiley, New York)

    MATH  Google Scholar 

  • Orszag, S.A. (1971): Stud. Appl. Math. 50, 293–327

    MathSciNet  MATH  Google Scholar 

  • Peyret, R. (1986): “Introduction to Spectral Methods”, Lect. Ser. 1986–04, Von Karman Inst. for Fluid Dynamics, Belgium

    Google Scholar 

  • Peyret, R., Taylor, T.D. (1983): Computational Methods for Fluid Flow, Springer Ser. Comput. Phys. (Springer, New York, Berlin, Heidelberg)

    Google Scholar 

  • Strang, G., Fix, G.J. (1973): An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, N.J.)

    MATH  Google Scholar 

  • Thomasset, F. (1981): Implementation of Finite Element Methods for Navier-Stokes Equations, Springer Ser. Comput. Phys. (Springer, Berlin, Heidelberg)

    Google Scholar 

  • Voigt, R.G., Gottlieb, D., Hussaini, M.Y. (1984): Spectral Methods for Partial Differential Equations (SIAM, Philadelphia)

    MATH  Google Scholar 

  • Zienkiewicz, O.C. (1977): The Finite Element Method, 3rd ed. (McGraw-Hill, London)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Centre for Advanced Numerical Computation in Engineering and Science — CANCES, The University of New South Wales, Sydney, 2052, Australia

    Professor Clive A. J. Fletcher

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  1. Professor Clive A. J. Fletcher
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© 1998 Springer-Verlag Berlin Heidelberg

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Fletcher, C.A.J. (1998). Weighted Residual Methods. In: Computational Techniques for Fluid Dynamics 1. Springer Series in Computational Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58229-5_5

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  • DOI: https://doi.org/10.1007/978-3-642-58229-5_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53058-9

  • Online ISBN: 978-3-642-58229-5

  • eBook Packages: Springer Book Archive

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